[FOM] Understanding Universal Quantification
Hartley Slater
slaterbh at cyllene.uwa.edu.au
Mon Feb 24 21:38:52 EST 2003
Allen Hazen writes: "Another [division amongst positions] is over the
question of whether the "understanding" of universal quantification
presupposes some sort of delimitation of the domain quantified over".
There is a technical result in this area which is not too well known,
which shows the epsilon calculus is not as subject to limitation as
the predicate calculus. See, for instance, the last section of my
entry 'Epsilon Calculi' in the Internet Encyclopedia of Philosophy.
Briefly:
A convenient form of the epsilon calculus arises simply by modifying
the predicate calculus truth trees, as found in, for instance,
Richard Jeffrey's 'Formal Logic: Its Scope and Limits', (McGraw-Hill,
New York, 1st Ed. 1967). Jeffrey has a rule of existential quantifier
elimination, (Ex)Fx |- Fa, in which 'a' must be new, and a rule of
universal quantifier elimination, (x)Fx |- Fb, in which 'b' must be
old, unless no other individual terms are available. Clearly, upon
adding epsilon terms to the language, the first of these rules can be
changed to: (Ex)Fx |- FexFx, (where 'e' is epsilon) while also the
two parts of the second rule can be replaced by the pair of rules:
(x)Fx |- Fex-Fx, Fex-Fx |- Fb (where 'b' is invariably old) to
produce an appropriate proof procedure.
But Jeffrey's rules only allow him 'limited upward correctness'
(Jeffrey 1967, p167), since he has to say, with respect to his
universal quantifier elimination rule, that the range of the
quantification there be limited merely to the universe of discourse
of the path below. This is because, if an initial sentence is false
in a valuation so also must be one of its conclusions. But the first
epsilon rule which replaces Jeffrey's rule ensures, instead, that
there is 'total upwards correctness'. For if it is false that
everything is F then, without any special interpretation of the
quantifier, one of the given consequences of the universal statement
is false, namely the immediate one, since Fex-Fx is in fact
equivalent to (x)Fx. A similar improvement also arises with the
existential quantifier elimination rule....
Not being able to specify the prime putative exception, 'ex-Fx', to
the universal statement '(x)Fx' leaves Jeffrey believing that there
must be a model for the quantifiers which restricts them to a certain
domain, which means that they do not necessarily range over
everything. But in the epsilon calculus the quantifiers do range over
everything, and there is no need to specify their range. This has
consequences for 'The Domain Principle' found in Cantor (Hallett, M.
'Cantorian Set Theory and Limitation of Size', Clarendon, Oxford,
1984 p25), and defended to the last by Priest in his attempt to
establish dialetheism ('Beyond the Limits of Thought' C.U.P. 1995).
I have discussed these further matters in Ch7 of my 'Logic Reformed'
(Peter Lang, Bern, 2002, ISBN 3-906768-57-0; US-ISBN 0-8204-5875-9).
--
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, School of Humanities
University of Western Australia
35 Stirling Highway
Crawley WA 6009, Australia
Ph: (08) 9380 1246 (W), 9386 4812 (H)
Fax: (08) 9380 1057
Url: http://www.arts.uwa.edu.au/PhilosWWW/Staff/slater.html
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